There are various tire measuring and/or tire testing operations that require the tire to be mounted in an inflated condition in order to carry out measuring and/or testing steps thereon. To permit such an operation to be performed, use may be made of an apparatus for mounting a pneumatic tire in a measuring and/or testing unit, the apparatus including first and second mounting rim portions which hold between them a pneumatic tire in an inflated condition. The apparatus has a guide for rotationally mounting the rim portions in concentric relationship with respect to an axis, together with a clamping device for non-rotatably clamping one of the rim portions to the guide along which that rim portion is displaceable in the released condition thereof. Such a tire mounting apparatus is used for example in measuring units for checking the uniformity of motor vehicle tires (in this respect reference may be directed to the company publication Holmann Report 89, imprint 09.84). The displaceable rim portion is guided slidably on the guide in the direction of the axis of rotation about which the tire is rotated, for example, for the purpose of setting different tire widths as measured at the radially inward circumference of the tire, that is to say at the tire beads. To produce such displacement of the displaceable rim portion, it is necessary to allow a certain guide play or clearance between the guide and the rim portion or a carrier on which the rim portion is fixed.